Nuclear matrix elements for the resonant neutrinoless double electron capture

The rate of the neutrinoless double electron capture ( 0n\ensuremath 0\nu ECEC) decay with a resonance condition depends sensitively on the mass difference between the initial and final nuclei of decay. This is where the JYFLTRAP Penning-trap measurements at the JYFL become invaluable in estimation of the half-lives of these decays. In this work the resonant 0n\ensuremath 0\nu ECEC decay is discussed from the point of view of its theoretical aspects, in particular regarding the resonance condition and the involved nuclear matrix elements (NME). The associated decay amplitudes are derived and the calculations of the NMEs by the microscopic many-body approach of the multiple-commutator model are outlined. The resonant 0n\ensuremath 0\nu ECEC decays of ⁷⁴Ge\ensuremath {\rm ^{74}Ge} and ¹³⁶Ce\ensuremath {\rm ^{136}Ce} are discussed as applications of the theory framework.